9709 · 6.3
Continuous random variables
Unlike discrete variables which have specific probabilities, continuous variables use a 'probability density function' (PDF) where probability is the area under the curve. We use integration to find these areas and other key measures like the mean and variance.
Need to know
What you need to know
- For $f(x)$ to be a valid PDF, it must satisfy two conditions.
- Condition 1: $f(x) \ge 0$ for all values of $x$. The graph can never be below the x-axis.
- Condition 2: The total area under the curve must be 1. $\int_{-\infty}^{\infty} f(x) dx = 1$. In practice, we integrate over the interval where $f(x)$ is non-zero.
Explanation
From Density to Probability
- The PDF, f(x), shows the relative likelihood of a value. It must be non-negative, and the total area under its curve over its entire range must equal 1.
- The CDF, F(x), gives the total probability up to a value x, P(X ≤ x). It's the cumulative area under the PDF. The probability between two points, P(a < X ≤ b), is simply the difference F(b) − F(a).
- The mean E(X) and variance Var(X) are found by integrating over the variable's range. E(X) is the integral of x⋅f(x), and Var(X) is found using the integral of x²⋅f(x).
- A common example is the uniform distribution, where the PDF is a constant height over a fixed interval. Its graph is a rectangle where height times width equals 1.